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Mirrors > Home > ILE Home > Th. List > cofunexg | GIF version |
Description: Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.) |
Ref | Expression |
---|---|
cofunexg | ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ∘ B) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 4762 | . . 3 ⊢ Rel (A ∘ B) | |
2 | relssdmrn 4784 | . . 3 ⊢ (Rel (A ∘ B) → (A ∘ B) ⊆ (dom (A ∘ B) × ran (A ∘ B))) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ (A ∘ B) ⊆ (dom (A ∘ B) × ran (A ∘ B)) |
4 | dmcoss 4544 | . . . . 5 ⊢ dom (A ∘ B) ⊆ dom B | |
5 | dmexg 4539 | . . . . 5 ⊢ (B ∈ 𝐶 → dom B ∈ V) | |
6 | ssexg 3887 | . . . . 5 ⊢ ((dom (A ∘ B) ⊆ dom B ∧ dom B ∈ V) → dom (A ∘ B) ∈ V) | |
7 | 4, 5, 6 | sylancr 393 | . . . 4 ⊢ (B ∈ 𝐶 → dom (A ∘ B) ∈ V) |
8 | 7 | adantl 262 | . . 3 ⊢ ((Fun A ∧ B ∈ 𝐶) → dom (A ∘ B) ∈ V) |
9 | rnco 4770 | . . . 4 ⊢ ran (A ∘ B) = ran (A ↾ ran B) | |
10 | rnexg 4540 | . . . . . 6 ⊢ (B ∈ 𝐶 → ran B ∈ V) | |
11 | resfunexg 5325 | . . . . . 6 ⊢ ((Fun A ∧ ran B ∈ V) → (A ↾ ran B) ∈ V) | |
12 | 10, 11 | sylan2 270 | . . . . 5 ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ ran B) ∈ V) |
13 | rnexg 4540 | . . . . 5 ⊢ ((A ↾ ran B) ∈ V → ran (A ↾ ran B) ∈ V) | |
14 | 12, 13 | syl 14 | . . . 4 ⊢ ((Fun A ∧ B ∈ 𝐶) → ran (A ↾ ran B) ∈ V) |
15 | 9, 14 | syl5eqel 2121 | . . 3 ⊢ ((Fun A ∧ B ∈ 𝐶) → ran (A ∘ B) ∈ V) |
16 | xpexg 4395 | . . 3 ⊢ ((dom (A ∘ B) ∈ V ∧ ran (A ∘ B) ∈ V) → (dom (A ∘ B) × ran (A ∘ B)) ∈ V) | |
17 | 8, 15, 16 | syl2anc 391 | . 2 ⊢ ((Fun A ∧ B ∈ 𝐶) → (dom (A ∘ B) × ran (A ∘ B)) ∈ V) |
18 | ssexg 3887 | . 2 ⊢ (((A ∘ B) ⊆ (dom (A ∘ B) × ran (A ∘ B)) ∧ (dom (A ∘ B) × ran (A ∘ B)) ∈ V) → (A ∘ B) ∈ V) | |
19 | 3, 17, 18 | sylancr 393 | 1 ⊢ ((Fun A ∧ B ∈ 𝐶) → (A ∘ B) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 × cxp 4286 dom cdm 4288 ran crn 4289 ↾ cres 4290 ∘ ccom 4292 Rel wrel 4293 Fun wfun 4839 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 |
This theorem is referenced by: cofunex2g 5681 |
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